Uniform consistency of nearest neighbour and kernel conditional Kaplan - Meier estimates* By Dorota M. Dabrowska Carnegie - Mellon University and University of California, Berkeley Technical Report No. 86 January 1987 *Research supported by the University of California Presidential Fellowship and by the National Institute of General Medical Sciences Grant SSSY iROl GM35416-02. Department of Statistics University of California Berkeley, California
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Dorota M. DabrowskaCarnegie - Mellon University andUniversity of California, Berkeley
Technical Report No. 86January 1987
*Research supported by the University of California Presidential Fellowship and by theNational Institute of General Medical Sciences Grant SSSY iROl GM35416-02.
Department of StatisticsUniversity of CaliforniaBerkeley, California
Uniform consistency of nearest neighbour
and kernel conditional Kaplan - Meier estimates*
Dorota M. Dabrowska
Carnegie - Mellon University and
University of California, Berkeley
Abstract
We consider a class of nonparametric regression estimates introduced by
Beran (1981) to estimate conditional survival functions in the presence of right
censoring. For both uncensored and censored case, we derive an analogue of the
Dvoretzky - Kiefer - Wolfowitz (1956) inequality for the tails of distributions of
kernel and nearest neighbour estimates of conditional survival function. This ine-
quality is next used to prove weak and strong uniform consistency results. The
developments rest on sharp exponential bounds for the oscillation modulus of
multivariate empirical processes obtained by Stute (1984a).
Key words and phrases: Kernel and nearest neighbour regression, right censoring,
oscillation modulus.
* Research supported by the U'niversity of California Presidential Fel-lowship and by the National Institute of General Ntedical Sciences GrantSSSY lROl GM35416-02.
1. Introduction.
Let T be a nonnegative random variable (rv) representing the survival time of an
individual taking part in a clinical trial or other experimental study, and let
Z = (Z1, * * , Zq) be a vector of covariates such as age, blood pressure, cholesterol
level. The survival time T is subject to right censoring so that the observable rv's are
given by Y = min (T ,X), D = I (T < X) and Z. Here X is a nonnegative rv represent-
ing times to withdrawal from the study. Denote by F (t I z) = P (T > t I Z = z),
Hl(t Iz) = P(Y > t, D = 1lZ = z) and H2(t Iz) =P(Y > t IZ = z) the respective con-
ditional survival functions and let
t
A(t I z) = -JF(s-I z)-1dF (s i z) (1.1)0
be the conditional cumulative hazard function associated with F (t z). In terms of the
cumulative hazard function we have
F (t I z) = exp {(Ac(t Iz)}FJl 1 - AA(s I z)}, (1.2)where AC (s I z) is the continuous component of A(s I z), the product is taken over the
set of discontinuities of A(s z) and s < t, and AA(s z) = A(s z) - A(s - I z). This is
the well known product-integral representation of distribution functions, see for
instance Peterson (1977), Gill (1980) and Beran (1981). To avoid trivial cases it is
assumed throughout that p = P (D = 1) satisfies 0 < p < 1. Furthermore, it is
assumed that T and X are conditionally independent given Z, which is a sufficient
condition to ensure identifiability of A(t I z) and F (t I z). Specifically, for any t such
t
that H2(t Iz)> 0 we have A(t Iz) = -J(H2(S-Iz))-1dH1(s Iz).0
Let (Yj, Dj, Zj), j = 1, ... ,n, be a sample of i.i.d. rv's each having the same
- 2 -
distribution as (Y,D,Z). Beran (1981) proposed to estimate the subdistribution func-n
nY I(Yj > t-)Bnj(z), where Bnj(z) is a random set of probability weights depending onj=1
the covariates only. Beran's estimates of A(t I z) and F (t I z) are provided by
t dHln(s jz)An (t I Z) = -J H2 (s z) (1.3)
and
Fn(t 1z) = H-fl - AAn(SI1)}, (1.4)where the product is taken over s < t. Both An (t z) and Fn (t I z) are right continuous
functions of t, jumps occur at discontinuity points of H in (t I z). Note that in the
homogeneous case, (1.3) and (1.4) are simply the Aalen - Nelson (Aalen (1978), Nel-
son (1972)) and Kaplan - Meier (1958) estimates. Hereafter, we refer to these esti-
mates as conditional Aalen - Nelson and conditional Kaplan - Meier estimates.
In this paper we consider two types of estimates. The first of them is a nearest
neighbour estimate of the form Hin (t z) = H3n (z-'Hin (t ,z) where
n
Hin(t,z) = (naq)-j z I(Y. > t, DG= - G
n ( 1.5)H2n (t,Z) = (nan,)f £ I(Yj > t)K (an n(G (z) - Gn (Zj)))
and
nH3n (z) = (naq)j1 Y K(a71(Gn (z) - Gn (Zj))) (1.6)
Here Gn (z) = (Gn 1(z1), , Gnq (Zq)) and Gnj (zj) is the empirical distribution func-
tion of Zij's, j = 1, . q. Further K is a kernel function in Rq and an is a
sequence of bandwidths. These estimates were discussed in particular by Stute (1984b,
1986a), Yang (1981) and Johnston (1981).
- 3 -
The second estimate is based on Nadaraya - Watson type weights and takes form
Hin (t jZ) = gn (Z) -Hin (t ,z) where
n
Hin (t,z) = (na I(Yj > t, Dj = -)K(ann (1.7)
H2n (t,z) = (nany-1 £ Ij(Y > t)K(an1(z - Z1))
and gn (z) is the Parzen - Rosenblatt (Parzen (1962), Rosenblatt (1971)) density esti-
mate
ngn (z) = (na,)jY z K(aq (z - Z1)) (1.8)
j=1These estimates were extensively studied in the literature, Nadaraya (1964), Watson
(1964), Rosenblatt (1969), Collomb (1981), Devroye (1981) and Mack and Silverman
(1982) are some references.
For uncensored data, conditional empirical processes were studied among others by
Stute (1986a & b) who considered pointwise consistency and weak convergence results
for estimates based on kernel and nearest neighbour weights. Horv'ath and Yandell
(1986) obtained functional laws of iterated logarithm and rates of Gaussian approxima-
tion. For censored data, the conditional Aalen - Nelson and Kaplan - Meier estimates
were discussed by Beran (1981) and Dabrowska (1987) who showed their pointwise
consistency and, respectively, weak convergence to a time transformed Brownian
motion.
In this paper we are concerned with uniform consistency results. For both uncen-
sored and censored case we develop an analogue of the Dvoretzky - Kiefer - Wol-
fowitz (1956) inequality providing an exponential bound for the tails of the distribution
of nearest neighbour and kernel estimates of the conditional survival function. This
bound is next used to establish weak and strong uniform consistency of these estima-
tors. The results rest on sharp exponential bounds for the oscillation modulus of
- 4 -
univariate and multivariate empirical processes obtained by Stute (1982, 1984b).
2. Main Results.
Nearest -neighbour and kernel estimates of the subdistribution functions Hi (t I z) are
discussed in Sections 2.1 and 2.2 respectively. The conditional Kaplan - Meier esti-
mates are considered in Section 2.3.
qIn what follows, for any rectangle I = Ili=1 [aj, bj ] in Rq, Is denotes a (small)
8-neighbourhood of I of the form
qIs = r1=1 [aj -8, bj + 8].
Unless specified otherwise, for any real function f (t ,z) = f (t ,z1 ..* , zq ) in R q+1
lIf Ill = sup {If (t ,z) 1: t ER, zEI}.
2.1. Nearest neighbour estimates. Let G denote the joint cdf of Zi and let Gi,j = 1, . , q be its j -th marginal cdf. Further, let
Co(u1 ... ,uq) = P (Gj (Zj) < uj;, i = 1, ,q). If G is a continuous cdf then CO
is a cdf with uniform marginal satisfying CO (G 1 (z 1), ,Gq (Zq)) = G (z1, XZq).
The following assumption on G will be needed.
A.1 The joint cdf G is continuous. The cdf CO has density co with respect to
qLebesgue measure on [ 0,1 ]q. If I = fIl=, [ a1, b1] is a rectangle contained in the
support of G, then for some 0 < 8 < 1,
0 <y= inf{co(u): uij } < sup {cO(u): uEIG } = r< , where IG is a 8-
qneighbourhood of G - IIj=i [Gj (aj ), Gj (bj)].
As for the kernel K, we assume that K is a density with finite support and has
bounded second partial derivatives. Under these assumptions K is function of
bounded variation in the sense of Hardy and Krause ( see e.g. Hildebrandt (1963),
- 5 -
Clarkson and Adams (1933)). Without loss of generality we assume
B.1 K is a density vanishing outside (-1, l)q. The first and the second partial
a _ a2derivatives K(J)(u) = K(u) and K01)(u) =- K(u) satisfy sup {IK(i)(u) I,aUJ Duj Du,IK(jl)(u)I: ue(_i,1)q, j,l = 1, ,q) <X < oo. The total variation of K is less
than X.
Our final assumption imposes growth rate conditions on an.
C.1 Let 0 < an < 1 and 0 < e < 1 satisfy 2 < -a q+2n(y/8X)min{1, l1(Fq2)},
eo > &1'2max{y/6kX, (712XF)1/2} and an < 6/(1 + 2e0), where e0 is some finite
number and 6, y, r, X are as is assumptions A. 1 and B. 1
Define Hin (t I z) = Hin (t,z) / H3n(z) where
Hin (t,z) = a-q JHi (t I u)K(a1 (G (z) -G (u)))dG (u), (2.1)H3n (z) = a- JK(a-l(G (z) - G (u)))dG (u), (2.2)
and G(u) = (G 1 (u 1) , Gq (Uq)). We shall decompose Hin (t I z) - Hin (t I z) into
two terms R in (t I z) and R2in (t I Z), i = 1,2. The exact definition of these terms is
given in Section 4. The following theorem provides exponential bounds for R lin (t,z)
and R 2in (t,z).
Theorem 2.1. If A.], B.] and C.1 hold then there exist constants di > 0,
i = 1, * , 4 (not depending on n, an, c, or the distribution of (Yi , Di, Z1)) such that
for i = 1,2
P (I Hin -Hin I1I >e-) <P(IRlin IIi >F/2)+P(I R2in III >e12)
P(IIR1in III > Fg/2) < dia-qexpf-d2d582na,q}P(IIR2in III > c1 2) < d3a-q7exp{-d4d6 na,'+l}
where d5 = nmin (X-2F, -2rF2) and d6 = min (d5, 1Yf71).
- 6 -
Theorem 2.1 can be used to derive uniform consistency results for the estimates
Hin (t I z). To handle the bias term, we assume
A.2. The functions co (u) and Hi (t I G-1(u)), G-1 (u) = (G _ 1 (u 1) , G 1 (uq)),
i = 1,2 have continuous first partial derivatives with respect to uj, uel and
supil a Hi(tIG-1(u))I<oo 0<infla cO(u)I < supi c0(u)l<oo where theauj ~~~~~~~~~~~au*
supremum and infimum extend over u EIG, t E R, i = 1,2 and j = 1, . , q.
Corollary 2.1. Suppose that the conditions A.] and B.] hold and an - 0.
(i) If nan+2 - then IIHin - Hin III ->p 0. (ii) If in addition
Ea, qexp{-pnag+q} < oo for all p > 0 then 1IHin - Hin 1-,> 0 a.s. (iii) If A.2 holds
then IIHin -Hi -I> 0.
Note that if naq+2 - o and logn Inaq+2 -* 0 then the assumption of (ii) is
satisfied. The following corollary deals with the rates of uniform convergence. To
handle the bias term we assume
A.3. The differentiability conditions of A.2 are satisfied by the second derivatives.
B.2. The kernel K satisfies ujuK(u)du = 0, j = 1, ,q.
Corollary 2.2. Suppose that the conditions A.1 and B.1 hold and let an -e 0 and
naq+2 __> 00 Set bn = logaqq/ (naq). (i) If bnan 2 - 0 then
IIH- -Hin II1 = OH(b$2). (ii) If in addition Ea 4 < oo for some 4 > 0 then
IInHi - Hin III = 0 (bn112) as n -e with probability 1. (iii) If A.3 and B.2 hold and
na1+4 - 0 then bn,112 tHin - Hi 1I1 -* 0.
2.2. Kernel estimates. In the case of kernel estimates we require the joint distribu-
tion of Zi's to have a density g with respect to Lebesgue measure. The marginal den-
sities Gj'are denoted by gj, j = 1, ,q. The cdf CO defined in Section 2.1 now
has density
- 7 -
co(u) = g (Gj71 (u1), ,Gq 1 (uq))IJ=1g1(G1 (uq)).The following assumptions will be needed.
A.4 The joint distribution G has density g with respect to Lebesgue measure. If
q
I =FJ=i [aj,bj1] is a rectangle contained in the support of g, then
O<y=inf{g(z), gj(zj):zeI8}<sup{g(z),gj(zj):zeIS}= r<oo for some 6-
neighbourhood of I. Moreover, 0 < 5F < 1 and sup{cO(u): uE IG,r = 1O <o
where I4r is a 6F-neighbourhood of GG - H1[G1(aj ), Gj (bj)].
As for the kernel K, we assume that K is a density with finite support and K is a
function of bounded variation in the sense of Hardy and Krause. Without loss of gen-
erality, assume
B.3. K is a density supported on (-1, l)q. K is a function of bounded variation
with total variation equal to X < oo.
Finally the following growth conditions on an and e will be needed
C.2. Let O< a < 1 and 0 < e < 1 satisfy 0 < an <6min(l, 1 /F), 4 <. nagqyXand e0 ' cy (2rorqX)-1 for some finite e0 > 0, where 6, y, F, ro and k are as in
assumptions A.4 and B.3.
Define Hin (t z) = Hin (t ,z) / g (z) where
Hn (t,z) = a~JqHi (t Iu)K(a, 1(z - u))dG(u) (2.3)
gn (z) = a q |K (an 1 (z - u)) dG (u) (2.4)
Theorem 2.2. Suppose that the conditions A.4, B.3 and C.2 hold. There exist
constants d7, d8 > 0 (not depending on n,anX, or the distribution of (Yi,Di, Zi)) such
that P (11 Hin - Hin II > E) < d7(Fan )exp {-d8dg _ nang where dg = y2/(F0Fq 2).
Similarly to the case of nearest neighbour estimates, we can derive uniform con-
sistency results. Assume
- 8 -
A.5. The functions g (z) and Hi (t z), i = 1,2, have continuous first partial deriva-
atives with respect to Z z E I and supi - Hi (t I z) < oo
aziO < inflag (z) I < supi -y g (z) i < oo where the sup and inf extend over t e R,
zE I8, i = 1,2 and j = 1, * * ,9q.
Corollary 2.3. Suppose that the conditions A.4 and B.3 hold and let an -4 0 and
na_- oc. (i) If a -expf- pnag}q- 0 for all p > 0 then IIHi, Hin 1l -,p 0. (ii)
If in addition I -q - pnaq I <oo for all p > 0 then 11 Hin - H Ill- 0 a.s. (iii)If A.S holds then 11 Hin - Hi IIz + 0.
Condition (i) is satisfied if naj+S - 0 for some c > 0. Condition (ii) holds if
log n Inaq _+ 0. The following corollary deals with the rates of uniform consistency.
To handle the bias term, assume
A.6. The differentiability conditions of A.5 are satisfied by the second derivatives.
Corollary 2.4. Suppose that the conditions A.4 and B.3 hold and let an -* 0 and
n- oo. (i) If bn = loga,/nann _< 0 then IHin - Hin III = Op (bn112) (ii) If in
addition Ea 4 <oo
for some 4 > 0 then IIHn - Hiin 11
= 0(b 1'2) as n -o with pro-
bability 1. (iii) If A.6 and B.2 hold and na q+4 then bnT1/211Hin - Hi II. - 0
2.3. Conditional Kaplan - Meier estimates. In what follows, for any 0 < t < 00
and any rectangle I in R q, if f (t,z) is a real function in Rq+1 then
lIf hIII = sup{If(t,Z)I: 0 < t < t, Z E I. To avoid problems with the tails of the
conditional subdistribution functions, we add the following assumptions.
A.7. For O< t < oo, inf{H2(t IG-1(u)): 0 < t < t, U IG} > 0 > 0, where I is
defined as in A.1, and G-1(u) = (G j1 (u 1),I -(u
A.8. For O < X < oo, inf{H2(t Iz): 0 < t < t, zeI} > 0 > 0, where I is defined as
-9-
in A.4.
The exponential bounds for the distribution of a suitably standardized version of
Fn (t I z) will be developed under the following assumptions on an and £.
C.3. Let 0 < an < 1 and 0 < c < 1 satisfy 2 < tO5na q+22(y/432X)min{1, 1/1Iq2},e0 ' e 12( /2)112max{y/(6XF), (y/2kr)112} and an < 6/(1 + 2e0), where e0 is
some finite number and 6,y,F,9 and X are as in assumptions A.1, A.7 and B.1.
C.4. Let 0< an < 1 and O< e < 1 satisfy 0 < an <6min(1, 1 /F),
216 < E05nafqy/X and e0 > 9y(roPqFX)-l/4 for some finite e0 > 0, where
6, y, 17,r0, 8 and X are as in assumptions A.4 and B.3.
Let Hin (t z) be defined as in Sections 2.1 and 2.2. Set
A~(ti) = ~dH1,n (s jz)An (t IZ)=-| ]o Hmn (s-Jz)
and
Fn (t Iz) = exp{-An(t IZ)}Hr (1-AAn (s IZ))s.t
where Anc is the continuous component of An the product is taken over the set of
discontinuities of An and AAn (s I z) = An (s zZ) - An (s - l z). The following theorem
provides exponential bounds for Fn (t IZ) - Fn (t I z). In the case of nearest neighbour
estimates, in analogy to Theorem 2.1, we use a decomposition of Fn (t z) - Fn (t I z)
into two terms Sln (t, z) and 52n (t, z). Their exact definitions are given in Section 6.
Theorem 2.3. (i) Nearest neighbour estimates: Suppose that the conditions A.1,
A.7, B.1 and C.3 hold. There exist constants ci > 0i = 1, , 4 such that
an (q+1) 5J I f3nj (zj) - Pnj (Uj ) I K(i)(a, -(G(z) - G(u)) co(G(u)) I Ei=l dGi (ui)
< 1 rqX max sup I a2nj (Gj (zj)) -2nj (vj)Iwhere the suprema extend over vj such that Gj (zj) - vj I < ean . Further, a2nj,
j = 1, ... ,q are the last q marginals of a2n. Lemma 3.1 applied with q1 = 0 q2 = 1
a = ean and s = Tin 112an l (Xrq ) entails P (IK(Q2ln ) IIB 2in 11>'i)
<qP(c(ea 1)> Tin1/2a, /(XEFq)) <qmle-la-lexp{-m2Ti2na / e (Xrq )2} where ml
and m2 are the constants of Lemma 3.1. The choice of 13 = qmle-1 and
14 = M2e-1q-2 completes the proof.
Lemma 4.3. Let A.] and B.] be satisfied. If 0 < an < /e and 0 < c < 1 satisfy
2<. naq(y/ 82) min{ 1, 1 / (q)} and e0 > ey/ (8XE) then there exist constants
d1,d2 > 0 such that for i = 1,2 P(I1Rlin III>g/2) < dla -qexp{ d2d5 qnaI} where
d5 = 7k-2 min(P1, 2).
Proof. We have 11 R lin III < I(Qln)y1 (11 B lin 1I + 11B 2in hII+ 1lB 12n 11, + 11 B 22n 11,). Under condition C. 1, the assumptions of Lemmas 4.1 and 4.2
are satisfied by Ti = vy/8. Furthermore
P (11 R in II > 6/ 2) < 2P (I (Qln)JIB lin 11 > vY/ 8) +2P(I(Q ) 11B 2in 11 > Ey/ 8) < 21 la, q exp{ 2-262 nanqy2 / (64X2r) )
+ 213 af exp- 1462 nany2/ (64%3E2)}.The conclusion follows by setting d1 = 2 max(11, 13) and d2 = min(12,14) / 64.
Lemma 4.4. Let A.] and B.] be satisfied. If 0 < an < 61e and 0 < Ti < 1 satisfy
2 < tin2aq+2 I (Erq2) and eO . 2fr / (XE) then there exist constants 15, 16>0 such that
- 17 -
for i = 1,2 P(I (Q21n)IIB3in 11 > 11) < l5an7exp{-166na,+l/(X)}
Proof. We have
I(Qln) IB3in(t Z) I
<21 ana +2) (Xrq 2) max sup I X2nj (Gj (zj)) - X2nj (vj) X2nil (GI (z1)) -X2nl (V1)
where the suprema extend over vj, v1 such that I Gj(zj) vj <ean and
IGi(zI) - v1 < ean, ],l = 1, ,q. Furthermore, 2nj j = 1, , q are the last q
marginals of X2n. Lemma 3.1 applied with q1 = 0, q2 = 1, a = ean and
< q 2m le -la jlexp{2m 2e-lrqna 1+ql (%Xq 2)} where m1 and m 2 are constants of
Lemma 3.1. The conclusion follows by setting 15 = q2mle1 and 16 = 2m2e1q 2*
Lemma 4.5. There exist constants 17, 18>0 such that P (Qfl) <
11exp{-126na q,l ,2/(6F)2}.
Proof. The Dvoretzky-Kiefer-Wolfowitz (1956) inequality entails
P(Qln) < P (11 Gnj- Gj1 >1/2a+l)/2/(6))j=1
< qm 1 expl-2 ena,+1q+/ (6XF)2}where m1 is the universal constant of the Dvoretzky-Kiefer-Wolfowitz inequality.
Lemma 4.6. Let A.] and B.] be satisfied. If 0 < an <6le and 0 < & < 1 satisfy
2 < gna~n+2(y/6X)min{l,l/(Fq2)} and eO . &l2max{y/(6XF),^y12/(32X)"2}, then
there exist constants 19, 110>0 such that P(Q2c) < lga -qexp{-110l11cna q+l} where
11 = min{y2I7-122, 72v2x-2 yr'X}-l
Proof. We have
P(Q2C) < P(sup(H3 -(z)-H3(z)) > infH3 -(z)- y/2, z E I)<P (11A3n III > PY12) < P (KQ1n) + p V(Qln) JIB 2n 11 > YF-1/ /6)+ P V (Qln) JIB22n 11 > 78 1/26) + P V (Qln)IIB32n 11 > rz/6)
The assumptions of Lemmas 4.1 and 4.2 are satisfied with q = y1/2 /6 and those of
- 18 -
Lemma 4.4 with rl = y /6. It follows that the right-hand side of the above inequality
is bounded by
7exp{-18 ena q+ / (6rX)2} + 11 a -qexp -12e na q I (36F2X2)}+ 13an exp{-14enan+/(6F)2} + 15a1 exp{-16ena+ y/ (617)}.
The conclusion follows by setting 19 = max(l 1, 13,15,17) and
110 = min(l2/36, 14/36, 1616, 18/36).
Lemma 4.7. Let A.] and B.1 be satisfied. If 0 < an < le and 0 < e < 1 satisfy
2 < £nan+2(y/6k)min{l,1/(Fq2)} and e0 > &'2max{y/(6XF), 71y21(2X%)"2}, then
there exist constants d3,d4>0 such that for i = 1,2
P(IIR2 Ii /2) <d2 qexp.-dq+ where d6 =
min{yP1X-2, y2F"2X-2 y,'1X-}.
Proof. We have IIR2inIIi < I(Qln n Q2n)Y1(IIB3in II + IIB33n III) +I(Qfc 'Q2c )IIHin -HiniI. Lemma 4.4 applied with rq = ey/4 and Lemmas 4.5 and
4.6 entail
P (IIR2in III > e/2) < 2P (IIB 3in ll (Q1,n) > uy/4) + P (Qlc) +1 (Q2c) < 215a71exp{ -l6 na,q'1y/(4rk)} +
l7expI{-l8naq+,1/(6Fk)2j + 19a -qexp{ -10l1enaq+l}.The conclusion follows by setting d 3 = max (215,17,1 9) and d 4 = min (16/4, 18/36, 1 11)
and d6 = 111.
To complete the proof of Theorem 2.1 it is enough to note that if C. 1 holds then
the conditions of Lemmas 4.3 and 4.7 are simultaneously satisfied. To prove Corol-
laries 2.1 and 2.2, we consider first the bias term.
Lemma 4.8. If A.2 holdts thien IIHin - Hi II, = 0 (an). If A.3 and B.2 hold then for
where v = (v 1, ,vq), -1< v < 1, j = 1, * * * , q. A one term Taylor expansion
shows that for i = 1,2 suplHin (t ,z) - H (t Iz)co(G(z))I = 0 (an). Since
H3n (z) = H2n(OZ) and H2(OIz)cO(G(z)) = co(G(z)), we also have
supIH3n (z) - cO(G(z))I = 0 (an). This implies the first part of the lemma. The second
follows from a two-term Taylor expansion.
Proof of Corollary 2.1. (i) For n sufficiently large, the conditions of Theorem 2.1
are satisfied. Since na 3 -> oo, we have a,-qexp{ pna+l} e* 0 for all p>O. It fol-
lows that IIHin - Hin I -p 0 as n -* oo. (ii) follows from (i) and the Borel - Cantelli
theorem. Lemma 4.8 implies (iii).
Proof of Corollary 2.2. (i) Choose arbitrary 4 >O and set
e = f-n = {bn ( + t)/d2d5l 1/2 For n sufficiently large, the conditions of Theorem 2.1
are satisfied. With this choice of E, we have P (IR uin Ii > ) = d a 4q e O so that
IIn I = Op (b1/2). Furthermore, P (hR 2in >R) .
d3exp {log a -q[ 1 - d (na q+2/log a-q)2]}, where d = d4(1 + 4)1"2I(d 2d5)112 Since
log a-qlna q+2 _* 0, for n sufficiently large P(hR2inqI>a) . -* 0O 0 that
hR2inhR = Op((b 12) as n -> oo. (ii) follows from (i) and the Borel - Cantelli theorem.
Lemma 4.8 implies (iii).
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5. Proof of Theorem 2.2 and Corollaries 2.3 and 2.4.
We consider first the processes Ain (t,z) = Hin (t,z) - Hi (t ,z), i = 1,2 where
Hin (t ,z) and Hin (t,z) are defined by (1.7) and (2.3).
Lemma 5.1. Let A.4 and B.3 be satisfied. If rj < 1 and an < 1 satisfy 2 < rlnanqlXand e0 2 r1(Fo1q7X)-l for some finite e0> 0, then there exist constants 11, 12 > 0 such
that for i = 1,2 P(1 Ain >A) . n naq
Proof. Recall Pin = Min - Mi, i = 1,2. Further, define
I (t ,z,u) = [t,o]xljL=1[min(uj,zj), max(uj,zj)]. Integration by parts (Hildebrandt
(1963)) yields, after some algebra
AinA(t,z)I = a jqlfI(y > t)K(a -1(z - u))dPin3(y,u)I (5.1)= aI Jf|in (I (t ,z;u)) dK (an (z-u))
where integration is restricted to those u-values for which Z- an < uj < zj + an,
j = 1, ,q. Set J (s,v;w) = s, 1]xIj?q1v[ Vj + min (O,wj ), vj + max (0,w )] and let
J0 be defined as in (3.2). If s = H2 (4*(t-)) and Vj = G1 (zj) then, after a change of
variable in the right-hand side of (5.1), we obtain from (3.2)
Ain (t ,z) I ' an q x1n (J0x J (s,v;w))i Idx(w) I if i = 1
naJi a2n (J (s ,v;w))i Id V(w)qI if i = 2.
Here integration is restricted to those wv-values for which
Since 1/R < a < /4, we can apply the exponential bound to both sums to obtain
P(:On(a) > s) < Rel'exp{-e2's2/a(1 + 6)21 +
Re 1'exp{-e2's262/(4a (1 + 6)2))< 2e,'exp{-e2's262/(4a(1 + 6)2))
provided 2 < sn 1/2, 32a < s262(1 - 26)2 and a < 6/4. Under assumptions of the
Lemma, since e1 and e2 are left unspecified, we can assume without loss of generality
that the last two growth conditions are satisfied. The conclusion follows by setting
e = 2e 1' and e2 = e2'62/(4(1 + 6)2).
For q2 = 2, the proof is similar to that of Theorems 1.5 and 1.7 in Stute (1984a).
An induction argument shows that the Lemma remains valid for arbitrary q2.
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TECHNICAL REPORTSStatistics Department
University of California, Berkeley
1. BREIMAN, L. and FREEDMAN, D. (Nov. 1981, revised Feb. 1982). How many variables should be entered in aregression equation? Jou. Assoc.. March 1983, 78, No. 381, 131-136.
2. BRILLINGER, D. R. (Jan. 1982). Some contrasting examples of the time and frequency domain approaches to time seriesanalysis. me Ihd Hyiosciences (A. H. El-Shaarawi and S. R. Esterby, eds.) Elsevier ScientificPublishing Co., Aisterdam, 1982, pp. 1-15.
3. DOKSUM, K. A. (Jan. 1982). On the performance of estimates in proportional hazard and log-linear models. SurvivalAnalysis. (John Crowley and Richard A. Johnson, eds.) IMS Lecture Notes - Monograph Series, (Shanti S. Gupta, seriesed.) 1982, 74-84.
4. BICKEL, P. J. and BREIMAN, L. (Feb. 1982). Sums of functions of nearest neighbor distances, moment bounds, limittheorems and a goodness of fit test. Ana.Qkb. Feb. 1982, 11. No. 1, 185-214.
5. BRILLINGER, D. R. and TUKEY, J. W. (March 1982). Spectrum estimation and system identification relying on aFourier transforn. e Colloected Worlc of J. W. Tukey. vol. 2, Wadsworth, 1985, 1001-1141.
6. BERAN, R. (May 1982). Jacklmife approximation to bootstrap estimates. Arm. S March 1984, 12 No. 1, 101-118.
7. BICKEL, P. J. and FREEDMAN, D. A. (June 1982). Bootstrapping regression models with many parameters.Lehmam FetsbrifLL (P. J. Bickel, K. Doksum and J. L. Hodges, Jr., eds.) Wadsworth Press, Belmont, 1983, 28-48.
8. BICKEL, P. J. and COLLINS, J. (March 1982). Minimizing Fisher information over mixtures of distributions. Sankhyi.1983, 45, Series A, Pt. 1, 119.
9. BREIMAN, L. and FRIEDMAN, J. (July 1982). Estimating optimal transformations for multiple regression and correlation.
10. FREEDMAN, D. A. and PETERS, S. (July 1982, revised Aug. 1983). Bootstrapping a regression equation: someempirical results. JASA. 1984, 79, 97-106.
11. EATON, M. L. and FREEDMAN, D. A. (Sept. 1982). A remark on adjusting for covariates in multiple regression.
12. BICKEL, P. J. (April 1982). Minimax estimation of the mean of a mean of a nonnal distribution subject to doing wellat a point. Recent Advances in Statistics, Academic Press, 1983.
14. FREEDMAN, D. A., ROTHENBERG, T. and SUTCH, R. (Oct. 1982). A review of a residential energy end use model.
15. BRILLINGER, D. and PREISLER, H. (Nov. 1982). Maximum likelihood estimation in a latent variable problem. Studiesin Econometric Tr Series. and Multivariate Statistics. (eds. S. Karlin, T. Amemiya, L. A. Goodman). AcademicPress, New York, 1983, pp. 31-65.
16. BICKEL, P. J. (Nov. 1982). Robust regression based on infinitesimal neighborhoods. Ann. Statist.. Dec. 1984, 12,1349-1368.
17. DRAPER, D. C. (Feb. 1983). Rank-based robust analysis of linear models. I. Exposition and review.
18. DRAPER, D. C. (Feb 1983). Rank-based robust inference in regression models with several observations per cell.
19. FREEDMAN, D. A. and FIENBERG, S. (Feb. 1983, revised April 1983). Statistics and the scientific method, Commentson and reactions to Freedman, A rejoinder to Fienberg's comments. Springer New York 1985 Cohort Analysis in SocialResearch, (W. M. Mason and S. E. Fienberg, eds.).
20. FREEDMAN, D. A. and PETERS, S. C. (March 1983, revised Jan. 1984). Using the bootstrap to evaluate forecastingequations. L. Qf Forecasting. 1985, Vol. 4, 251-262.
21. FREEDMAN, D. A. and PETERS, S. C. (March 1983, revised Aug. 1983). Bootstrapping an econometric model: someempirical results. IBES. 1985, 2, 150-158.
22. FREEDMAN, D. A. (March 1983). Structural-equation models: a case study.
23. DAGGEIT, R. S. and FREEDMAN, D. (April 1983, revised Sept. 1983). Econometrics and the law: a case study in theproof of antitrust damages. Prc.of the Berkeley Conference. in honor of Jerzy Neyman and Jack Kiefer. Vol I pp.123-172. (L. Le Cam, R. Olshen eds.) Wadsworth, 1985.
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24. DOKSUM, K. and YANDELL, B. (April 1983). Tests for exponentiality. Handbook of ,Statisics (P. R. Krishnaiah andP. K. Sen, eds.) 4, 1984.
25. FREEDMAN, D. A. (May 1983). Comments on a paper by Markus.
26. FREEDMAN, D. (Oct. 1983, revised March 1984). On bootstrapping two-stage least-squares estimates in stationary linearmodels. Am. Sttis. 1984, 12, 827-842.
27. DOKSUM, K. A. (Dec. 1983). An extension of partial likelihood methods for proportional hazard models to generaltransformation models. Ann. S 1987, 15, 325-345.
28. BICKEL, P. J., GOETZE, F. and VAN ZWET, W. R. (Jan. 1984). A simple analysis of third order efficiency of estimateof the Neyman-Kiefer Conference. (L. Le Cam, ed.) Wadsworth, 1985.
29. BICKEL, P. J. and FREEDMAN, D. A. Asymptotic normality and the bootstrap in stratified sampling. Ann. Statist._12 470-482.
30. FREEDMAN, D. A. (Jan. 1984). The mean vs. the median: a case study in 4-R Act litigation. JBES. 1985 Vol 3pp. 1-13.
31. STONE, C. J. (Feb. 1984). An asymptotically optimal window selection rule for kemel density estimates. Ann. Statist..Dec. 1984, 12, 1285-1297.
32. BREIMAN, L. (May 1984). Nail finders, edifices, and Oz.
33. STONE, C. J. (Oct. 1984). Additive regression and other nonparametric models. Ann. Statist.. 1985, 13, 689-705.
34. STONE, C. J. (June 1984). An asymptotically optimal histogram selection rule. Proc. of the Berkeley Conf. in Honor ofJerzy Neyman and Jack Kiefer (L. Le Cam and R. A. Olshen, eds.), II, 513-520.
35. FREEDMAN, D. A. and NAVIDI, W. C. (Sept. 1984, revised Jan. 1985). Regression models for adjusting the 1980Census. Statistical Science. Feb 1986, Vol. 1, No. 1, 3-39.
36. FREEDMAN, D. A. (Sept. 1984, revised Nov. 1984). De Finetti's theorem in continuous time.
37. DIACONIS, P. and FREEDMAN, D. (Oct. 1984). An elementary proof of Stirling's formula. Amer. Math Monthly. Feb1986, Vol. 93, No. 2, 123-125.
38. LE CAM, L. (Nov. 1984). Sur l'approximation de familles de mesures par des familles Gaussiennes. Ann. Inst.Henri Poincare. 1985, 21, 225-287.
39. DIACONIS, P. and FREEDMAN, D. A. (Nov. 1984). A note on weak star uniformities.
40. BREIMAN, L. and IHAKA, R. (Dec. 1984). Nonlinear discriminant analysis via SCALING and ACE.
41. STONE, C. J. (Jan. 1985). The dimensionality reduction principle for generalized additive models.
42. LE CAM, L. (Jan. 1985). On the normal approximation for sums of independent variables.
43. BICKEL, P. J. and YAHAV, J. A. (1985). On estimating the number of unseen species: how many executions werethere?
44. BRILLINGER, D. R. (1985). The natural variability of vital rates and associated statistics. Biometrics, to appear.
45. BRILLINGER, D. R. (1985). Fourier inference: some methods for the analysis of array and nonGaussian series data.Water Resources Bulletin. 1985, 21, 743-756.
46. BREIMAN, L. and STONE, C. J. (1985). Broad spectrum estimates and confidence intervals for tail quantiles.
47. DABROWSKA, D. M. and DOKSUM, K. A. (1985, revised March 1987). Partial likelihood in transformation modelswith censored data.
48. HAYCOCK, K. A. and BRILLINGER, D. R. (November 1985). LIBDRB: A subroutine library for elementary timeseries analysis.
49. BRILLINGER, D. R. (October 1985). Fitting cosines: some procedures and some physical examples. Joshi Festschrift.1986. D. Reidel.
50. BRILLINGER, D. R. (November 1985). What do seismology and neurophysiology have in common? - Statistics!Comptes Rendus Math. Rep. Acad. Sci. Canada. January, 1986.
51. COX, D. D. and O'SULLIVAN, F. (October 1985). Analysis of penalized likelihood-type estimators with application togeneralized smoothing in Sobolev Spaces.
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52. O'SULLIVAN, F. (November 1985). A practical perspective on ill-posed inverse problems: A review with somenew developments. To appear in JornaQfStat1stical Science,
53. LE CAM, L. and YANG, G. L. (November 1985, revised March 1987). On the preservation of local asymptotic normalityunder information loss.
54. BLACKWELL D. (November 1985). Approximate normality of large products.
55. FREEDMAN, D. A. (June 1987). As others see us: A case study in path analysis. Lunin!of EducationalStatistis, 12, 101-128.
56. LE CAM, L. and YANG, G. L. (January 1986). Replaced by No. 68.
57. LE CAM, L. (February 1986). On the Bernstein - von Mises theorem.
58. O'SULLIVAN, F. (January 1986). Estimation of Densities and Hazards by the Method of Penalized likelihood.
59. ALDOUS, D. and DIACONIS, P. (February 1986). Strong Uniform Times and Finite Random WaLks.
60. ALDOUS, D. (March 1986). On the Markov Chain simulation Method for Uniforn Combinatorial Distributions andSimulated Annealing.
61. CHENG, C-S. (April 1986). An Optimization Problem with Applications to Optimal Design Theory.
62. CHENG, C-S., MAJUMDAR, D., STUFKEN, J. & TURE, T. E. (May 1986, revised Jan 1987). Optimal step typedesign for comparing test treatments with a control.
63. CHENG, C-S. (May 1986, revised Jan. 1987). An Application of the Kiefer-Wolfowitz Equivalence Theorem.
64. O'SULLIVAN, F. (May 1986). Nonparametric Estimation in the Cox Proportional Hazards Model.
65. ALDOUS, D. (JUNE 1986). Finite-Time Implications of Relaxation Times for Stochastically Monotone Processes.
66. PITMAN, J. (JULY 1986, revised November 1986). Stationary Excursions.
67. DABROWSKA, D. and DOKSUM, K. (July 1986, revised November 1986). Estimates and confidence intervals formedian and mean life in the proportional hazard model with censored data.
68. LE CAM, L. and YANG, G.L. (July 1986). Distinguished Statistics, Loss of information and a theorem of Robert B.Davies (Fourth edition).
69. STONE, C.J. (July 1986). Asymptotic properties of logspline density estimation.
71. BICKEL, P.J. and YAHAV, J.A. (July 1986). Richardson Extrapolation and the Bootstrap.
72. LEHMANN, E.L. (July 1986). Statistics - an overview.
73. STONE, C.J. (August 1986). A nonparametric framework for statistical modelling.
74. BIANE, PH. and YOR, M. (August 1986). A relation between Levy's stochastic area formula, Legendre polynomial,and some continued fractions of Gauss.
76. O'SULLIVAN, F. (September 1986). Relative risk estimation.
77. O'SULLIVAN, F. (September 1986). Deconvolution of episodic hormone data.
78. PITMAN, J. & YOR, M. (September 1987). Further asymptotic laws of planar Brownian motion.
79. FREEDMAN, D.A. & ZEISEL, H. (November 1986). From mouse to man: The quantitative assessment of cancer risks.To appear in Statistical Science.
80. BRILLINGER, D.R. (October 1986). Maximum likelihood analysis of spike trains of interacting nerve cells.
81. DABROWSKA, D.M. (November 1986). Nonparametric regression with censored survival time data.
82. DOKSUM, K.J. and LO, A.Y. (November 1986). Consistent and robust Bayes Procedures for Location based onPartial Information.
83. DABROWSKA, D.M., DOKSUM, KA. and MIURA, R. (November 1986). Rank estimates in a class of semiparametrictwo-sample models.
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84. BRILLINGER, D. (December 1986). Some statistical methods for random process data from seismology andneurophysiology.
85. DIACONIS, P. and FREEDMAN, D. (December 1986). A dozen de Finetti-style results in search of a theory.Ana. Ins. Hemri Poincare. 1987, 23, 397-423.
86. DABROWSKA, D.M. (January 1987). Unifonn consistency of nearest neighbour and kemel conditional Kaplan- Meier estimates.
87. FREEDMAN, DA., NAVIDI, W. and PETERS, S.C. (February 1987). On the impact of variable selection infitting regression equations.
88. ALDOUS, D. (February 1987, revised April 1987). Hashing with linear probing, under non-uniform probabilities.
89. DABROWSKA, D.M. and DOKSUM, K.A. (March 1987, revised January 1988). Estimating and testing in a twosample generalized odds rate model.
90. DABROWSKA, D.M. (March 1987). Rank tests for matched pair experiments with censored data.
91. DIACONIS, P and FREEDMAN, D.A. (April 1988). Conditional limit theorems for exponential families and finiteversions of de Finetti's theorem. To appear in the Journal of Applied Probability.
92. DABROWSKA, D.M. (April 1987, revised September 1987). Kaplan-Meier estimate on the plane.
92a. ALDOUS, D. (April 1987). The Harmonic mean formula for probabilities of Unions: Applications to sparse randomgraphs.
93. DABROWSKA, D.M. (June 1987, revised Feb 1988). Nonparametric quantile regression with censored data.
94. DONOHO, D.L. & STARK, P.B. (June 1987). Uncertainty principles and signal recovery.
95. RIZZARDL F. (Aug 1987). Two-Sample t-tests where one population SD is known.
96. BRILLINGER, D.R. (June 1987). Some examples of the statistical analysis of seismological data. To appear inProceedings, Centennial Anniversary Symposium, Seismographic Stations, University of California, Berkeley.
97. FREEDMAN, DA. and NAVIDI, W. (June 1987). On the multi-stage model for carcinogenesis. To appear inEnvironmental Health Perspectives.
98. O'SULLIVAN, F. and WONG, T. (June 1987). Determining a function diffusion coefficient in the heat equation.99. O'SULLIVAN, F. (June 1987). Constrained non-linear regularization with application to some system identification
problems.
100. LE CAM, L. (July 1987, revised Nov 1987). On the standard asymptotic confidence ellipsoids of Wald.
101. DONOHO, D.L. and LIU, R.C. (July 1987). Pathologies of some minimum distance estimators.
102. BRILLINGER, D.R., DOWNING, K.H. and GLAESER, R.M. (July 1987). Some statistical aspects of low-doseelectron imaging of crystals.
103. LE CAM, L. (August 1987). Harald Cramer and sums of independent random variables.
104. DONOHO, A.W., DONOHO, D.L. and GASKO, M. (August 1987). Macspin: Dynamic graphics on a desktopcomputer.
105. DONOHO, D.L. and LIU, R.C. (August 1987). On minimax estimation of linear functionals.
106. DABROWSKA, D.M. (August 1987). Kaplan-Meier estimate on the plane: weak convergence, LIL and the bootstrap.
107. CHENG, C-S. (August 1987). Some orthogonal main-effect plans for asymmetrical factorials.
108. CHENG, C-S. and JACROUX, M. (August 1987). On the construction of trend-free run orders of two-level factorialdesigns.
109. KLASS, M.J. (August 1987). Maximizing E max S / ES': A prophet inequality for sums of I.I.D. mean zero variates.
110. DONOHO, D.L. and LIU, R.C. (August 1987). The "automatic" robustness of minimum distance functionals.
111. BICKEL, P.J. and GHOSH, J.K. (August 1987). A decomposition for the likelihood ratio statistic and the Bartlettcorrection - a Bayesian argument.
112. BURDZY, K., PITMAN, J.W. and YOR, M. (September 1987). Some asymptotic laws for crossings and excursions.
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113. ADHIKARL A. and PriMAN, J. (September 1987). The shortest planar arc of width 1.
114. R1TOV, Y. (September 1987). Estimation in a linear regression model with censored data.
115. BICKEL, PJ. and RITOV, Y. (September 1987). Large sample theory of estimation in biased sampling regressionmodels I.
116. RITOV, Y. and BICKEL, P.J. (September 1987). Unachievable information bounds in non and semiparametric models.
117. RITOV, Y. (October 1987). On the convergence of a maximal correlation algorithm with alternating projections.
118. ALDOUS, D.J. (October 1987). Meeting times for independent Markov chains.
119. HESSE, C.H. (October 1987). An asymptotic expansion for the mean of the passage-time distribution of integratedBrownian Motion.
120. DONOHO, D. and LIU, R. (October 1987, revised March 1988). Geometrizing rates of convergence, II.
121. BRILLINGER, D.R. (October 1987). Estimating the chances of large earthquakes by radiocarbon dating and statisticalmodelling. To appear in Statistics a Guide to the Unknown.
122. ALDOUS, D., FLANNERY, B. and PALACIOS, J.L. (November 1987). Two applications of urn processes: The fringeanalysis of search trees and the simulation of quasi-stationary distributions of Markov chains.
123. DONOHO, D.L. and MACGIBBON, B. (November 1987). Minimax risk for hyperrectangles.
124. ALDOUS, D. (November 1987). Stopping times and tightness II.
125. HESSE, C.H. (November 1987). The present state of a stochastic model for sedimentation.
126. DALANG, R.C. (December 1987). Optimal stopping of two-parameter processes on hyperfinite probability spaces.
127. Same as No. 133.
128. DONOHO, D. and GASKO, M. (December 1987). Multivariate generalizations of the median and timmed mean II.
129. SMITH, D.L. (December 1987). Exponential bounds in Vapnik-tervonenkis classes of index 1.
132. HESSE, C.H. (December 1987). A Bahadur - Type representation for empirical quantiles of a large class of stationary,possibly infinite - variance, linear processes
133. DONOHO, D.L. and GASKO, M. (December 1987). Multivariate generalizations of the median and trimmed mean, I.
134. DUBINS, L.E. and SCHWARZ, G. (December 1987). A sharp inequality for martingales and stopping-times.
135. FREEDMAN, DA. and NAVIDI, W. (December 1987). On the risk of lung cancer for ex-smokers.
136. LE CAM, L. (January 1988). On some stochastic models of the effects of radiation on cell survival.
137. DIACONIS, P. and FREEDMAN, D.A. (April 1988). On the uniform consistency of Bayes estimates for multinomialprobabilities.
137a. DONOHO, D.L. and LIU, R.C. (1987). Geometrizing rates of convergence, I.
138. DONOHO, D.L. and LIU, R.C. (January 1988). Geometrizing rates of convergence, HI.
139. BERAN, R. (January 1988). Refining simultaneous confidence sets.
140. HESSE, C.H. (December 1987). Numerical and statistical aspects of neural networks.
141. BRILLINGER, D.R. (January 1988). Two reports on trend analysis: a) An Elementary Trend Analysis of Rio NegroLevels at Manaus, 1903-1985 b) Consistent Detection of a Monotonic Trend Superposed on a Stationary Time Series
142. DONOHO, D.L. (Jan. 1985, revised Jan. 1988). One-sided inference about functionals of a density.
143. DALANG, R.C. (February 1988). Randomization in the two-armed bandit problem.144. DABROWSKA, D.M., DOKSUM, K.A. and SONG, J.K. (February 1988). Graphical comparisons of cumulative hazards
for two populations.
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145. ALDOUS, D.J. (February 1988). Lower bounds for covering times for reversible Markov Chains and random walks ongraphs.
146. BICKEL, P.J. and RITOV, Y. (February 1988). Estimating integrated squared density derivatives.
147. STARK, P.B. (March 1988). Strict bounds and applications.
148. DONOHO, D.L. and STARK, P.B. (March 1988). Rearrangements and smoothing.
149. NOLAN, D. (March 1988). Asymptotics for a multivariate location estimator.
150. SEILLIER, F. (March 1988). Sequential probability forecasts and the probability integral transform.
151. NOLAN, D. (March 1988). Limit theorems for a random convex set.
152. DIACONIS, P. and FREEDMAN, D.A. (April 1988). On a theorem of Kuchler and Lauritzen.
153. DIACONIS, P. and FREEDMAN, DA. (April 1988). On the problem of types.
154. DOKSUM, KA. (May 1988). On the correspondence between models in binary regression analysis and survival analysis.
155. LEHMANN, E.L. (May 1988). Jerzy Neyman, 1894-1981.
156. ALDOUS, D.J. (May 1988). Stein's method in a two-dimensional coverage problem.
157. FAN, J. (June 1988). On the optimal rates of convergence for nonparametric deconvolution problem.
158. DABROWSKA, D. (June 1988). Signed-rank tests for censored matched pairs.
160. BERAN, R.J. and MILLAR, P.W. (June 1988). Tests of fit for logistic models.
161. BREIMAN, L. and PETERS, S. (June 1988). Comparing automatic bivariate smoothness.
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